Step 1 Make the sauce. Heat the olive oil in a large, heavy-bottomed pot over medium heat. Add the onion, garlic and anchovies and season with salt and pepper. Cook, stirring occasionally, until the onion is totally softened and translucent (without letting it brown), 8-10 minutes. Add the tomato paste if using and continue to cook, stirring, until the tomato paste has turned a deeper brick red color, about 2 minutes.
Step 2 Using your hands, crush the whole tomatoes into smaller, bite-sized pieces and add them and the crushed tomatoes to the pot, stirring to scrape up any bits from the bottom. Fill one of the tomato cans halfway with water, swirl it around to get all the bits in the can and add it to the pot. Season with salt and pepper. Bring to a simmer and cook, stirring occasionally, until the tomato sauce has thickened and flavors have come together, 35-45 minutes. You want it to be as thick as tomato sauce from a jar — any looser and the lasagna will be too wet to cut into nice pieces.
Step 3 Preheat the oven to 425°F and set a large pot of salted water to boil.
Step 4 Assemble the lasagna. Set aside 1 cup mozzarella. In a medium bowl, combine the remaining mozzarella, the ricotta, Parmesan, cream, and grated garlic clove; season with salt and pepper.
Step 5 Cook the lasagna noodles in the boiling water until just softened (before al dente), 4 to 5 minutes. Drain and separate any noodles that are trying to stick together, drizzling them with a bit of olive oil to prevent them from sticking further.
Step 6 Spoon a bit of sauce on the bottom of a 3-quart baking dish and top with a layer of noodles, avoiding any heavy overlap (some overlap is fine and inevitable). Top with about 1¼ cups of sauce and dollop one-fourth of the cheese mixture over. Top with another layer of noodles and repeat three more times, ending with the last of the noodles (depending on size of the noodle/shape of the baking dish, you may have a few extra noodles) and the last of the sauce. Top with the reserved 1 cup mozzarella and more Parmesan, if you like.
Step 7 Cover loosely with aluminum foil and place the baking dish on a foil-lined rimmed baking sheet (to prevent any overflow from burning on the bottom of your oven). Bake until the pasta is completely tender and cooked through and the sauce is bubbling up around the edges, 25-30 minutes. Remove the foil and continue to bake until the lasagna is golden brown on top with frilly, crispy edges and corners, another 35-45 minutes. Let cool slightly before eating.
